{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(X, \\Sigma, \\mu)$ be a [[measure space|P15VQS5DJ9]].\nLet $f \\colon X \\to \\R$ or $f \\colon X \\to \\C$ be a [[measurable function|YYNCJSR0PJ]].\nThe function $f$ is *integrable*, or *Lebesgue integrable*, if\n$$\\int_X \\abs{f} \\, d\\mu \u003c \\infty,$$\nwhere the left side is the [[integral of the non-negative function|DG76G5Z6XW]] $\\abs{f}$.\n\n## Remarks\n\nExample: on $\\R$ with Lebesgue measure, $f(x) = e^{-\\abs{x}}$ is measurable since it is continuous.\nIts integral is $\\int_\\R e^{-\\abs{x}} \\, dx = 2 \u003c \\infty$, so $f$ is integrable.\n\nNon-example: $g(x) = \\sin(x) / x$ on $[1, \\infty)$ with Lebesgue measure is measurable but not integrable.\nOn each interval $[k\\pi, (k+1)\\pi]$ the integral of $\\abs{g}$ is at least $2 / ((k+1)\\pi)$, and these bounds have an infinite sum.\nIts improper Riemann integral still converges, so improper integrals and Lebesgue integrals differ here.\n\nA real function $f$ is integrable if and only if both $f^+ = \\max(f, 0)$ and $f^- = \\max(-f, 0)$ have finite integral.\nA function with values in $[-\\infty, \\infty]$ is called integrable under the same condition, and it is then finite almost everywhere.\nThe integrable functions form a vector space, often written $\\mathcal{L}^1(\\mu)$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "X7GTF3D98F",
    "kind": "definition",
    "msc": [
      "28A25",
      "26A42"
    ],
    "name": "Integrable function",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lebesgue_integral"
      }
    ],
    "tag": "RSAK8YHMXF",
    "uses": [
      "P15VQS5DJ9",
      "YYNCJSR0PJ",
      "DG76G5Z6XW"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/RSAK8YHMXF/",
  "links": [
    {
      "field": "defines",
      "target": "X7GTF3D98F"
    },
    {
      "field": "uses",
      "target": "P15VQS5DJ9"
    },
    {
      "field": "uses",
      "target": "YYNCJSR0PJ"
    },
    {
      "field": "uses",
      "target": "DG76G5Z6XW"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/RSAK8YHMXF.md",
  "msc": [
    "28A25",
    "26A42"
  ],
  "name": "Integrable function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/RSA/RSAK8YHMXF.md",
  "status": null,
  "tag": "RSAK8YHMXF",
  "uri": "m2://RSAK8YHMXF",
  "url": "https://math2.tamnd.com/wiki/integrable-function-definition/",
  "writing": {
    "tag": "RSAK8YHMXF",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}